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Let me refer to the example here. Suppose $X$ is a birth-death (BD) process (represents population size) that evolves by:

$X \to X+1$ if a birth occurs with rate $\mu$,

$X \to X-1$ if a death occurs with rate $\theta$.

Suppose $T_n$ is the first passage time of a BD process from state $n$ to state $0$. Then: $$T_n = T + \frac{\mu}{\mu + \theta}T_{n+1} + \frac{\theta}{\mu + \theta}T_{n-1}$$

where $T$ is a random variable for the time needed to get out of state $n$, and $T ～ \exp(\mu + \theta)$.

Now I would like to take the Laplace transform of both sides. The accepted answer is as follows:

$$ \phi_n(s) = \frac{\mu + \theta}{\mu + \theta + s} \left( \frac{\mu}{\mu + \theta} \, \phi_{n+1}(s) + \frac{\theta}{\mu + \theta} \, \phi_{n-1}(s) \right)$$

My question is: If $T_{n},T,T_{n-1},T_{n+1}$ are all random variables, why does the Laplace transform for the sum $\frac{\mu}{\mu + \theta}T_{n+1} + \frac{\theta}{\mu + \theta}T_{n-1}$ apply the linear rule, while it is not the case with that sum and $T$?

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  • $\begingroup$ If I understand correctly, with probability $\mu/(\mu+\theta)$, $T_n$ is equal in law to $T + T_{n+1}$, and otherwise it is equal to $T + T_{n-1}$, with $T$, $T_{n-1}$ and $T_{n+1}$ independent. Is that correct? $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 10, 2020 at 14:07
  • $\begingroup$ @MateuszKwaśnicki: I think you may be right. I am just still wondering: with probability $\frac{\mu}{\mu + \theta}$, will $T$ remain to follow $exp(\mu+\theta)$, or would that be the case we KNOW that it would go to state $n+1$ and $T$ should be following $exp(\mu)$ now? $\endgroup$
    – user36706
    Commented Oct 10, 2020 at 14:27
  • $\begingroup$ Wait first, then toss a coin. $T$ has exponential distribution with parameter $\mu+\theta$, only after waiting for $T$ units of time the particle decides where to go. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 10, 2020 at 14:29
  • $\begingroup$ Thank you a million! That really clarifies the intuition. Just one more question: is there any specific rule for applying Laplace transform to summation like this, or is there any decent textbook about this topic where I can find similar problem? (You are welcomed to make it as a complete answer too...) $\endgroup$
    – user36706
    Commented Oct 10, 2020 at 14:42
  • $\begingroup$ I suppose Chapter 14 of Feller's An Introduction to Probability Theory and Its Applications, vol. 2 is a standard reference. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 10, 2020 at 18:23

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